Proof (sketch)

If $\mathbf{2}$ is Sierpinski space (two points $0$, $1$ and three opens $\emptyset$, $\{1\}$, and $\{0, 1\}$), then the continuous map $I = [0, 1] \to \mathbf{2}$ taking $0$ to $0$ and $t \gt 0$ to $1$ is a weak homotopy equivalence¹.

The essential construction in the proof is as follows: for any finite topological space $X$ with specialization order $\mathcal{O}(X)$, the topological interval map $I \to \mathbf{2}$ induces a weak homotopy equivalence $B\mathcal{O}(X) \to X$:

(where we implicitly identify $\Delta^{op}$ with the category $Int$ of finite intervals with distinct top and bottom, so that $[n] \mapsto Int([n], I)$ is a covariant functor on $\Delta$). A few remarks on this construction:

• The interval $[n]$ has $n+2$ elements, two of which are the distinct top and bottom. The space $Int([n], I)$ is the $n$-dimensional affine simplex. The space $Int([n], \mathbf{2})$ has $n+1$ points $0, 1, \ldots, n$, where $j$ is in the closure of $j+1$ for $0 \leq j \lt n$. The map $Int([n], I) \to Int([n], \mathbf{2})$ induced by $I \to \mathbf{2}$ takes every interior point of $Int([n], I)$ to $n \in Int([n], \mathbf{2})$.

• Informally, the isomorphism on the right says that any finite topological space $X$ can be constructed by gluing together copies of Sierpinski space $\mathbf{2}$, just as any preorder can be constructed by gluing together copies of the preorder $\{0 \leq 1\}$. More formally, the isomorphism is established for objects $X$ in the equivalent category $PreOrd_{fin}$, by restricting an isomorphism over objects $X$ of the larger category $PreOrd$, given by the counit of a nerve and realization adjunction

  $$\int^{[n] \in \Delta} Cat([n], X) \cdot Int([n], \{0 \leq 1\}) \cong \int^{[n] \in \Delta} Cat([n], X) \cdot [n] \stackrel{counit}{\cong} X$$

  where the counit is an isomorphism because the inclusions $PreOrd \hookrightarrow Cat \stackrel{nerve}{\hookrightarrow} Set^{\Delta^{op}}$ are fully faithful.

On the other hand, any finite simplicial complex $K$ is homotopy equivalent to its barycentric subdivision. This is $B P K$, the geometric realization of the nerve of the poset $P K$ whose elements are simplices ordered by inclusion. Thus finite posets model the weak homotopy types of finite simplicial complexes.